Download The Cavendish Experiment in General Relativity

arXiv:gr-qc/9709063v1 24 Sep 1997
The Cavendish Experiment in General Relativity∗
Dieter Brill
Department of Physics
University of Maryland
College Park MD 20742, USA
Abstract
Solutions of Einstein’s equations are discussed in which the “gravitational force” is balanced by an electrical force, and which can serve as
models for the Cavendish experiment.
1
Introduction
One of many useful lessons one can learn from Engelbert is the appreciation of
simple situations and examples that nonetheless can teach us valuable physics.
For me, such an Engelbert lesson was an introduction to the Bertotti-Robinson
universe (which, as Engelbert added with characteristic precision that extends
also to the history of physics, was first discovered by Levi-Civita), and its relation to extremal solutions [1]. Below is a bit of physics that we can learn from
extremal solutions to Einstein’s equations.
In General Relativity there is a well-defined sense in which the equations of
motion for particles follow from the field equations. This is well known but not
easily checked out,1 for the necessary manipulations are rather formidable. It
has been remarked [2] that when predictions of General Relativity are based on
particle equations of motion they appear to lack the transparency and cogency
that we appreciate in Newtonian physics and in some alternative theories; that
even the outcome of the Cavendish experiment has not been derived in a way
that is both simple and rigorous; and that, at least in the case of two-dimensional
(“planar”) translational symmetry there exist “anti-Cavendish” solutions of Einstein’s field equations, describing slabs that do not attract each other. (In these
solutions there are no other interactions than gravity between the slabs, but the
stress-energy of the matter is “exotic.”)
∗ Contribution to Festschrift volume for Engelbert Schucking, to be published by Springer
Verlag
1 I really mean nachvollziehbar, a fashionable German word that seems to have no good
English equivalent.
1
In the present contribution we examine a question suggested by these considerations2 , namely whether there are simple models in General Relativity that are
relevant to the Cavendish experiment. We will construct one such model that
is easily analyzed and whose predictions agree with the expected experimental
outcome. These models are not confined to the plane symmetric case for which
they were first discussed, and they have no connection with the “exotic” slab
solutions. Nevertheless we begin with a few elementary remarks about the
special status of planar symmetry in General Relativity as compared to other
field theories.
2
Planar symmetry
In electrostatics, problems with planar symmetry (such as two parallel charged
plates) are among the simplest to treat. The translational and rotational symmetry of the physical setup prevents dependence of physical quantities on the
transverse (y, z) directions. The problem therefore becomes one-dimensional.
Physically one cannot, of course, realize strict translational symmetry, because
the system’s total charge and mass would be infinite. However, one can approximate the one-dimensional situation by systems whose properties are independent
of y and z out to some large distance D, when one considers only longitudinal
distances x small compared to D. In the limit D → ∞ the one-dimensional
approximation becomes arbitrarily accurate, and reasonable physical quantities
have finite limits. These include the electric field, the force per area, and the
acceleration of the plates. (However, when the total charge on the plates is nonzero, the electrostatic potential does not have a finite limit, if it is normalized
to zero at infinity.)
Another simple feature of translationally symmetric electrostatics is the
uniqueness of the relative acceleration between the plates. (The electric field is
unique up to an additive constant.) If one has any solution with the appropriate
symmetry, it is the correct solution.3 This simplicity makes the (approximately)
parallel plate geometry so useful in both pedagogy and practice.
In Newtonian gravitation the situation is essentially identical; everything
one knows about electrostatics can be taken over (with the appropriate sign of
the force), except that there is no arbitrary charge to mass ratio — the (strong)
principle of equivalence fixes the ratio of gravitational to inertial mass to be a
positive constant. One might expect that the simplicity of the parallel plate
2I
thank Prof. C. Alley (University of Maryland) for numerous discussions which called
attention to the status of the Cavendish experiment vis a vis General Relativity, and in which
he supplied the experimental ideas mentioned in passing below.
3 This is true provided the plates are indeed static. In a typical experiment one balances
the electric force between plates by an elastic force, and measures how much elastic force is
needed to keep the plates static. If plates of finite size and non-negligible charge were allowed
to accelerate, they would of course radiate. The radiation reaction would affect the plates’
net acceleration, and this would depend on radiation conditions imposed at infinity.
2
geometry will also carry over in General Relativity.
There are of course important differences between these theories, which can
destroy the analogy. One relevant difference that is usually cited is the role of
the potential. In electrostatics and in Newtonian gravity the potential has no
direct physical meaning, separate from the fields. In general relativity the analogous quantity is the metric, and it measures directly the physically meaningful
space-time distances. When the size D of the system increases (with constant
mass density), the Newtonian potential, normalized to zero at infinity, typically
diverges. This is not a serious problem in electrostatics or in Newtonian gravity, because another normalization can be chosen with impunity. However, a
diverging metric offers more serious problems, and is certainly not allowed in
an asymptotically flat spacetime. On the other hand, it is not obvious that
this divergence cannot be undone in the limit by suitable gauge changes; and in
any case one can take the view that if translationally symmetric solutions exist,
they should (approximately) describe physically realistic parallel plates, since
the General Relativity solutions for finite plates presumably exist.
Static solutions with plane symmetry have in fact been studied in general
relativity, for example by Taub [3] for matter with a fluid equation of state.
Unfortunately they do not readily lend themselves to physical interpretation of
the type sought here. (One can however show on the basis of this work that, as
expected, no solution exists with vanishing pressure and positive mass density.)
Also, the relation of these solutions to any description of finite parallel plates
with proper (asymptotically flat) behavior at infinity is not transparent.
3
Exact, static solutions
Static solutions would not seem to present a very versatile arena for exploring
the features of the gravitational interaction. They do however correspond to
a possible physical arrangement that would reasonably be used in a sensitive
experiment to measure the strength of the gravitational interaction (G). In
the usual Cavendish experiment,4 even if initially the proof mass is in free fall,
the long-time behavior is typically governed by an interplay of gravitational
interaction and torsion fiber reaction. The initial acceleration can generally not
be measured as accurately as the final displacement, which one can model as
masses with constant separation — in other words, a static situation.
4 Prof.
Alley (private communication) points out that this could be modified to realize
the plane-symmetric geometry by replacing the usual masses with parallel plates, one of
which is suspended (for example, by means of the traditional torsion fiber) so that the total
force on it can be monitored. This geometry has several advantages, for example that the
distance between the plates does not have to be known with great accuracy, and that many
of the devices used in a parallel plate electrostatic measurement to increase the accuracy,
such as “guard rings” to make the field more uniform, could be adapted to the gravitational
experiment. However, I do not know of any attempt to obtain a more accurate measurement
of G in this way.
3
The gravitational interaction is then measured by the force necessary to keep
the masses apart, and the basic nature of this force is electrical. (One could
replace the force of the torsion fiber by the explicitly electrical force obtained,
for example in the parallel plate version of the experiment, by putting equal
charges on the plates.) We model this force by assuming that each volume
also carries a net charge, proportional to the mass of that volume, and all of
the same sign. For a suitable choice of the constant charge/mass ratio, the
attractive gravitational and repulsive electrical forces will then balance in the
Newtonian description.
How do we describe this situation in General Relativity? Because mass
and charge is present we must solve the Einstein and the Maxwell equations,
as well as the equations of motion of the matter. The source in the Einstein
equations is the stress-energy of the electric field and that of the matter; the
source in the Maxwell equations is the charge density of the matter; and these
equations imply the matter equation of motion, at least for the simplest kind of
matter, “charged dust”.5 If it is indeed possible to balance the forces in detail
— an expectation discouraged by the nonlinear nature of Einstein gravitation,
but encouraged by the absence of interaction energies in the the corresponding
Newtonian situation — then there should be a static solution. It is a remarkable
theorem [4] that such solutions not only exist, but that the fields have a unique
form under these conditions, the Majumdar-Papapetrou form [5] that is wellknown when gravity is generated not by matter but only by charged black holes
(and the stress-energy of their electric field). In the latter case the geometry
and field can represent any static arrangement of a finite number of black holes
with an extremal charge.
The Majumdar-Papapetrou ansatz for the metric and field can be written
as
Aµ = V δtµ
(1)
ds2 = −V −2 dt2 + V 2 dx2 + dy 2 + dz 2
By explicit computation6 of the Einstein tensor Gαβ and the Maxwell stressEM
one finds agreement of most of the components of the two,
energy tensor Tαβ
for example
EM
Gxx = V −2 −Vx2 + Vy2 + Vz2 = Txx
EM
Gxy = −2Vx Vy = Txy
etc.
5 The static sources in these solutions are unstressed, due to the detailed balance between
electric and gravitational forces. So we can imagine that these are elastic bodies rather than
dust, but with vanishing stress and strain. The stress-energy tensor is then the same as that
of dust, and the solution still applies. It is clear that in this model the stress-energy tensor
satisfies all energy conditions one might reasonably want to impose.
6 I am grateful to the group of Prof. C. Alley for providing me with many of the results
cited below, as obtained by their computer calculations. The conclusions drawn from these
calculations are my own and have not been fully discussed with Prof. Alley.
4
The exception is (α, β) = (t, t). Similarly one finds that the Aµ of Eq (1.1)
satisfy most of the components of the vacuum Maxwell equations,
Fαβ ;β = 0 except for α = t.
This structure of the field equations is appropriate for a static dust source,
since that type of source contributes only to the components that are excepted
above. For these one has the condition (in units where the gravitational and
electromagnetic coupling is unity)
Gtt − TttEM
Ftβ ;β
=
=
−2V −5 ∇2 V = Tttmatter
−V −4 ∇2 V = Jtmatter .
(2)
Here the Laplacian ∇2 is to be evaluated in the flat three-dimensional background metric dx2 + dy 2 + dz 2 .
All the equations are satisfied if charged dust can supply both sources in the
equations (1.2). For this type of matter, with mass density ρ and charge density
σ, we have
matter
Tαβ
=
ρuα uβ
Jαmatter
=
σuα .
¿From the metric (1.1) we find that the unit four-velocity for the static matter
has the form uα = V −1 δαt . Thus we see that equations (1.2) are satisfied if we
choose
ρ = σ = V −3 ∇2 V.
(3)
The equation relating the “potential” V and the source ρ is very similar to
the Newtonian equation; in the vacuum region they are identical. One way to
make a correspondence between the two is the following: Given any Newtonian potential VN (vanishing at infinity) and source ρN one finds a solution of
equation (1.3) by
V = 1 + VN
ρ = ρN /V 3 .
Thus ρ has the same support as ρN , and the two differ only slightly if the
gravitational fields are weak, |VN | ≪ 1.
This solution to the equations of general relativity has all the physically
reasonable properties one expects; but could there be other solutions to the
same problem with different properties? Suppose any static solution to the
Einstein-Maxwell equations is given. For simplicity, confine attention to the
region outside the matter. Let the electrostatic potential At of the solution be
V . Let gij be the spacelike metric on the three-dimensional hypersurfaces that
are orthogonal to the timelike Killing vector. One can then show that when
one modifies the metric by a conformal factor V −2 , its Ricci tensor vanishes,
Rij [V −2 gij ] = 0. The three-dimensional modified metric must therefore be flat,
and hence the original metric and field must have the form of Eq (1.1). In this
sense, then, the solutions given above are unique.
5
4
Test particle motion
It is not necessary to verify separately that the equations of motion for the
matter are satisfied by the solution given by equations (1.1, 1.3), because the
matter motion is a consequence of the field equations. However, as a further
check that this solution is reasonable, we derive the equation of motion for a
test particle in the fields of this solution.
As in Newtonian physics, the general relativistic motion of test particles in
the general metric (1.1) (even with V harmonic) is not integrable, but there is
always an energy integral. The energy integral is enough to find the motion if
we know that it is confined to one coordinate line. This is the case for example
when there is planar (x, y) or axial (about the z-axis) symmetry. We therefore
confine attention to these cases where the energy integral yields the essential
information about the motion.
In the case of uncharged particles we can apply the usual theorems about
geodesics, that any Killing vector like ∂/∂t yields a conservation law of the
corresponding covariant component of the 4-velocity u, ut = −E. From the
metric (1.1) we therefore have, with τ = proper time
E = −gtt
1 dt
dt
= 2 .
dτ
V dτ
We also know that setting the length of u to unity is always an integral of the
equation of motion,
1
u · u = −1 = − 2
V
2
dt
dτ
+
1
= E2.
V2
+V
2
dx
dτ
2
.
(4)
Elimination of dt/dτ yields
dx
dτ
2
(5)
So for geodesic motion the quantity 1/V 2 acts as an effective potential, and the
particle will be deflected from the Killing orbit (x, y, z) = const by an amount
proportional to ∇V , as in the Newtonian description.
For weak fields we have V ≈ 1 − Φ where Φ is the Newtonian potential (for
a spherically symmetric mass M , Φ = −M/r), hence
1
2
dx
dτ
2
+ Φ ≈ 21 (E 2 − 1),
which is (essentially) the usual energy integral for potential motion, yielding the
usual motion corresponding to attractive gravity.
6
When the particle is charged (with charge/mass q) the corresponding conservation laws are derived from the variational principle
Z
δ (u · u + 2qu · A)dτ = 0.
If there is a Killing vector that leaves the metric and the potential A invariant,
there is a conserved quantity (Noether’s theorem); for the Killing vector ∂/∂t
the conserved quantity is the momentum conjugate to t, −E = ut + qAt , or
E=
1 dt
q
− .
V 2 dτ
V
We substitute this in (1.4), eliminate dt/dτ , and find
dx
dτ
2
+
1 − q2
2Eq
+
= E2.
V2
V
We see that the attractive gravitational potential of the comparable equation
(1.5) is reduced, and becomes repulsive for q 2 > 1. There is also a contribution
to the potential that is proportional to E.
In the special (“extremal”7) case q 2 = 1 and E = 0, an initially static test
particle remains at rest at any position (as does the matter that produces these
fields). This is the case where attractive gravity and repulsive electrostatics
are in perfect balance. If the particle is initially not quite at rest, then Eq
must be slightly negative (since V > 0), hence the term 2Eq/V represents
an attractive potential. It may be interpreted as a response of the particle’s
increased “relativistic mass” to gravity, with no compensating increase in the
particle’s charge.
5
Conclusion
We have exhibited the unique static solutions to the Einstein-Maxwell-matter
field equations that represent an arbitrary distribution of extremally charged
matter in the form of dust. In particular, these solutions can be a good approximation to the geometry of a Cavendish experiment. Because all the charges
have the same sign, the electric interaction is repulsive between two volumes
of matter. The constancy in time of the physical distance between the masses
implies, in ordinary language, a balancing attractive gravitational interaction.
In this sense we have shown that in general relativity, as in Newtonian gravity,
the gravitational interaction between the bodies is nonzero and attractive. Because the solution is valid only when the charge has the extremal value, such
7 In this context the term “extremal” is somewhat misleading: it is the maximum charge
that a black hole of the given mass could have, but is not a large charge at all for that mass
of ordinary matter to carry.
7
a balancing Cavendish experiment could be used to find the extremal charge
value for a given mass.8 Measurement of this extremal charge to mass ratio is
equivalent to measuring the gravitational constant G.
References
[1] E. Schucking, personal communication.
[2] C. Alley, p. 464 in Fundamental Problems in Quantum Theory, Ann. New
York Acad. Sci. 755 (1995); p. 125 in M. Barone and F. Selleri (Eds.),
Frontiers of Fundamental Physics, Plenum Press (1994).
[3] A. H. Taub, Phys. Rev. 103, 454 (1956); p. 133 in L. O’Raifeartaigh (Ed.),
General Relativity, Oxford (1972)
[4] P. Chruściel and N. S. Nadirashvili, Class. Quant. Grav., 12, L17 (1995)
also see P. Chruściel, Contemporary Mathematics, 170, 23 (1994) and the
references cited there.
[5] S. D. Majumdar, Phys. Rev. 72, 390 (1947); A. Papapetrou, Proc. Roy.
Irish Acad. A51, 191 (1947)
[6] M.F. Fitzgerald and T.R Armstrong, IEEE Trans. on Instrumentation and
Measurement 44, 494 (1995).
8 Results from an electrically balancing Cavendish experiment have recently been reported
[6]; however, that experiment did not measure the extremal charge value because (for good
reasons) the attracting “large mass” was not the same as the repelling electrode.
8